Dynamic person-position matching decision method based on hesitant fuzzy number information

In view of the fact that people pay more and more attention to the principle of "getting the position according to the person" and "adapting the person to the position" in person-position matching, a dynamic person-position matching decision method based on hesitant fuzzy numbers is proposed. First, the dynamic person-position matching problem with hesitant fuzzy numbers is described. Then, according to hesitant fuzzy evaluation matrices of positions and candidates, expected score matrices of bilateral subjects are calculated. Furthermore, based on the idea of the generalized optimal order method and the dominant correlation and the missing correlation coefficients, satisfaction means of people and positions are calculated. According to satisfaction means, growth satisfactions at each period are obtained, and then the exponential decay formula is used to determine weights of growth satisfactions. Dynamic satisfactions of bilateral subjects are calculated by aggregating initial satisfaction means and growth satisfactions. On this basis, a stable person-position matching model considering dynamic satisfactions is established and then is solved to obtain the optimal stable person-position matching scheme. Finally, the feasibility and effectiveness of the proposed method are verified by an example analysis of person-position matching. Main contributions of this paper are as follows: an effective calculation method for the missing correlation coefficient is presented; a novel effective calculation method for dynamic satisfactions is proposed by introducing the correlation parameter to combine the missing correlation coefficient with the dominant correlation coefficient; an effective stable person-position matching model considering dynamic satisfactions is established.

1.A dynamic person-position matching decision method based on hesitant fuzzy numbers is proposed.2. The missing correlation coefficient is proposed to provide a basis for the calculation of satisfactions of bilateral subjects.3. A novel calculation method for initial satisfactions of bilateral subjects that combines the dominant and missing correlation coefficients is designed.4. Growth satisfactions are calculated to facilitate bilateral subjects to understand the status of each stage and improve their behaviors in time. 5.A stable person-position matching model considering dynamic satisfactions is established, which provides a more reasonable solution to the problem of person-position matching in the promotion process.
The remaining structure of this paper is as follows: "Preparatory knowledge" section introduces concepts of hesitant fuzzy numbers and stable person-position matching."Dynamic person-position matching decision under a hesitant fuzzy environment" section discusses the dynamic person-position matching decision under a hesitant fuzzy environment."Example analysis" section verifies the effectiveness and feasibility of the proposed method through a case of person-position matching."Conclusions" section summarizes this paper.
Definition 2 (Ref. 26) i=1 } be a HFN, and the number of elements in the HFN is represented by l h ; then its score function s(h) is expressed as:

Person-position matching and stable person-position matching
In person-position matching, let the position set be A = {A 1 , A 2 , A 3 , ..., A f } and the candidate set be B = {B 1 , B 2 , B 3 , ..., B g } , where A i is the i-th position in A and B j is the j-th candidate in B , i ∈ F={1, 2, 3, ..., f } , j ∈ G={1, 2, 3, ..., g} , f ≤ g.Definition 3 (Ref. 27) For a mapping then ∂ is called a person-position matching, where ∂(A i ) = B j means that A i matches with B j and ∂(B j ) = B j means that B j does not match in ∂.
Definition 4 (Ref. 28) For a mapping ∂:A ∪ B → A ∪ B , α ij is the satisfaction of the manager of position A i to candidate B j and β ij is the satisfaction of candidate B j to position A i .If there are not the following situations: then ∂ is called a stable person-position matching.

Problem description
Aiming at the dynamic person-position matching problem under a hesitant fuzzy environment, the position set is , where a a ij (t k ) represents the a -th hesitant fuzzy evaluation value of the manager of position A i to candidate The problems studied in this paper is to obtain the optimal dynamic person-position matching scheme according to hesitant fuzzy evaluation matrices.The solution idea is as follows: First, expected score matrices of bilateral subjects are calculated according to hesitant fuzzy evaluation matrices of positions and candidates.Then dynamic satisfactions of bilateral subjects are calculated by two correlation coefficients and exponential decay formula.On this basis, a stable person-position matching model is established and solved to obtain the optimal person-position matching scheme.The flow chart of dynamic person-position matching decision under a hesitant fuzzy environment is shown in Fig. 1.

Calculation of dynamic satisfactions of candidates and positions
According to hesitant fuzzy evaluation matrices In the following, the generalized priority method 29 is used to calculate dynamic satisfactions of people and positions by introducing step length, generalized series and two preference relations.First, the definition of step length based on expected scores s a ij (t k ) and s b ij (t k ) is given.Definition 5 (Ref. 28) Let the expected score s a ij (t k ) of the manager of position A i to candidate B j at time t k be divided into x + 1 levels, that is, the evaluation value of the manager of position A i at time t k is at most superior (2) www.nature.com/scientificreports/ to or inferior to that of candidate B j x levels; then the step length of the manager of position A i to candidate B j at time t k is expressed by γ A ij (t k ) , which is calculated as follows: Then the expected score s b ij (t k ) of candidate B j to position A i at time t k is also divided into x + 1 levels, and the step length of candidate B j to position A i at time t k is expressed by γ B ij (t k ) , which is calculated as follows: Then, the definition of generalized series is introduced based on the step length.
Definition 6 (Ref. 28) Assume that expected scores of candidates B j and B j ′ with respect to position A i at time t k are s a ij (t k ) and s a ij ′ (t k ) ; then the level of two candidates B j and B j ′ obtained by comparison is called the generalized series, expressed as r ijj ′ (t k ) , which is calculated as follows: Assuming that expected scores of managers of positions A i and A i ′ with respect to candidate B j at time t k are s b ij (t k ) and s b i ′ j (t k ) , respectively, r ii ′ j (t k ) is calculated as follows: In Eq. ( 6), r ijj ′ (t k ) indicates the rank of expected scores given by candidates B j and B j ′ with respect to position A i at time t k ; s a ij (t k ) ≻ s a ij ′ (t k ) is understood that the expected score of candidate B j is r ijj ′ (t k ) levels better than that of candidate B j ′ with respect to position A i at time t k ; s a ij (t k ) ≺ s a ij ′ (t k ) is understood that the expected score of candidate B j is r ijj ′ (t k ) levels worse than that of candidate B j ′ with respect to position A i at time t k ; s a ij (t k ) ≈ s a ij ′ (t k ) is understood that the expected score of candidate B j is equal to that of candidate B j ′ with respect to position A i at time t k .s a ij (t k ) ?s a ij ′ (t k ) is understood that the level of the expected score of candidates B j and B j ′ with respect to position A i at time t k is missing.The meaning of r ii ′ j (t k ) in Eq. ( 7) is similar to that of r ijj ′ (t k ) in Eq. (6).
Next, the generalized priority method is used to transform expected scores into satisfaction means of managers of positions and candidates.Definition 7 (Ref. 30) Assume that expected scores are divided into x + 1 levels; {≻, ≺, ≈, ?} denotes the set of preference relations, which can be understood as {better, worse, same, missing}, S, S ′ ∈ {≻, ≺, ≈, ?} , and d(S, S ′ ) represents the distance between score preference relations S and S ′ .Then wherein, x ≻ max denotes the maximum among distances between preference relations S and x ≻ of the dominant score, and x ≻ min denotes the minimum among distances between preference relations S and x ≻ of the dominant score.
Remark 1 According to literature 29 , distances among expected score preference relations are shown in Table 2, where a is the independent variable, satisfying a > 0.
According to literature 29 , the dominant correlation coefficient of the score preference relation η x ≻ (S ′ ) is calculated as follows: x , (5) x .
(6) Definition 8 Assume that expected scores are divided into x + 1 levels, score preference relations are S, S ′ ∈ {≻, ≺, ≈, ?} , and d(S, S ′ ) represents the distance between score preference relations S and S ′ .Then wherein, ?max denotes the maximum among distances between preference relations S and ? of the missing score and ?min denotes the minimum among distances between preference relations S and ? of the missing score.
On this basis, the missing correlation coefficient of the score preference relation η ?(S ′ ) is calculated as follows: where δ ∈ [0, 1] is the resolution coefficient, and usually satisfies δ = 0.5.By Eq. ( 8) and Table 2, the dominant correlation coefficient is calculated as follows: The generalized equal series r ijj ′ (t k ) is substituted into Eq. ( 10) and the relative satisfaction α ≻ ijj ′ (t k ) of the manager of position A i to candidate B j based on the dominant correlation coefficient at time t k is calculated as follows: Similarly, by Eq. ( 9) and Table 2, the missing correlation coefficient is calculated as follows: The generalized equal series r ijj ′ (t k ) is substituted into Eq.( 12), and the relative satisfaction α ?ijj ′ (t k ) of the manager of position A i to candidate B j based on the missing correlation coefficient at time t k is calculated as follows: x ≺, ≈, ?}}, x ≺, ≈, ?}}.
(9) η ?(S ′ ) = � ?min +δ� ?max d(S, ?) + δ� ?max , www.nature.com/scientificreports/Furthermore, the correlation parameter θ(0 ≤ θ ≤ 1 ) is introduced to calculate the satisfaction mean of the manager of position A i to candidate B j at time t k , i.e., By Eq. ( 14), the satisfaction mean matrix �(t k ) = [α ij (t k )] f ×g of side A to side B at time t k is obtained.Then, considering the change of growth satisfaction means, the growth satisfaction mean matrix ��(t k ) of side A to side B at time t k is calculated according to satisfaction mean matrix �(t k ) as follows: Similarly, the generalized equal series r ii ′ j (t k ) is substituted into Eq. ( 10) and the relative satisfaction β ≻ ii ′ j (t k ) of candidate B j to position A i based on the dominant correlation coefficient at time t k is calculated as follows: The generalized equal series r ii ′ j (t k ) is substituted into Eq.( 12), and the relative satisfaction β ?
ii ′ j (t k ) of candidate B j to position A i based on the missing correlation coefficient at time t k is calculated as follows: Furthermore, the correlation parameter θ(0 ≤ θ ≤ 1 ) is introduced to calculate the satisfaction mean of candidate B j to position A i at time t k , i.e., By Eq. ( 18), the satisfaction mean matrix �(t k ) = [β ij (t k )] f ×g of side B to side A at time t k is obtained.Then, considering the change of growth satisfaction means, the growth satisfaction mean matrix According to literature 30 , the exponential decay formula is used to determine growth satisfaction weights of bilateral subjects at time t k , i.e., where, ρ(0 ≤ ρ ≤ 1 ) is the attenuation coefficient, which reflects the change of the importance of growth satisfactions over time.
Finally, according to the initial satisfaction matrix �(t 1 ) , the growth satisfaction matrix ��(t k ) and weights ω � 2 , ..., ω � t k , ..., ω � q , the dynamic satisfaction matrix ] f ×g of side A to side B is obtained as follows: where www.nature.com/scientificreports/According to the initial satisfaction matrix �(t 1 ) , the growth satisfaction matrix ��(t k ) and weights ω � 2 , ..., ω � t k , ..., ω � q , the dynamic satisfaction matrix ] f ×g of side B to side A is obtained as follows:

Establishment of stable person-position matching model considering dynamic satisfactions
First, assume that the stable person-position matching matrix is

Solution of stable person-position matching model considering dynamic satisfactions
For objective functions D A and D B , considering that dimensions of where, ω 1 and ω 2 are weights of objective functions D A and D B , respectively, satisfying 0 ≤ ω 1 , ω 2 ≤ 1 ,

Decision steps for dynamic person-position matching under a hesitant fuzzy environment
For the dynamic person-position matching problem under a hesitant fuzzy environment, steps of the proposed decision method are as follows: www.nature.com/scientificreports/ Step 1.According to hesitant fuzzy evaluation matrices ] f ×g are calculated by Eqs. ( 2) and (3) respectively.Step 2.
According to expected score matrices ) of bilateral subjects are calculated at time t k by Eqs. ( 4) and ( 5).On this basis, generalized series r ijj ′ (t k ) and r ii ′ j (t k ) between bilateral subjects are calculated at time t k by Eqs. ( 6) and ( 7).
Relative satisfactions α ≻ ijj ′ (t k ) and α ?ijj ′ (t k ) are obtained at time t k by substituting the generalized series r ijj ′ (t k ) into Eqs.( 10) and ( 12); then the satisfaction mean matrix �(t k ) = [α ij (t k )] f ×g at time t k is established by Eq. ( 14).
According to the satisfaction mean matrix �(t k ) , the growth satisfaction mean matrix ��(t k ) = [�α ij (t k )] f ×g at time t k is calculated by Eq. ( 15).Step 5.
Relative satisfactions β ≻ ii ′ j (t k ) and β ?ii ′ j (t k ) are obtained at time t k by substituting the generalized series r ii ′ j (t k ) into Eqs.( 10) and ( 12); then the satisfaction mean matrix �(t k ) = [β ij (t k )] f ×g at time t k is established by Eq. ( 18).
According to the satisfaction mean matrix �(t k ) , the growth satisfaction mean matrix ��(t k ) = [�β ij (t k )] f ×g of side B to side A is calculated at time t k by Eq. ( 19).Step 7.
According to the initial satisfaction matrix �(t 1 ) , the growth satisfaction matrix ��(t k ) and weights ω � 2 , ..., ω � t k , ..., ω � q , the dynamic satisfaction matrix is solved by the relevant software to obtain the optimal stable person-position matching scheme.

Sensitivity analysis
To illustrate the effectiveness of the method proposed in this paper, the person-position matching problem will be solved from many aspects, and obtained optimal matching schemes are compared.(1) In the case of different correlation parameters, person-position matching models without considering stability constraints are similarly established, and person-position matching schemes in various situations are obtained by solving models, as shown in Table 20.(2) In the case of different correlation parameters θ , person-position matching models considering stability constraints are established, and stable person-position matching schemes in various situations are obtained by solving models, as shown in Table 21.(3) In the case of different weights ω 1 and ω 2 , person-position matching models considering stability constraints are established, and stable person-position matching schemes in various situations are obtained by solving models, as shown in Table 22.
It can be seen from Tables 20 and 21 that when different correlation parameters θ are used, there are some differences among optimal person-position matching schemes obtained by whether considering stability constraints or not.For example, when θ = 0 , the optimal person-position matching scheme  Compared with existing methods, main innovations of this paper are as follows: (1) The calculation method for the missing correlation coefficient is proposed.( 2) By calculating growth satisfactions, the status of bilateral subjects in each time period is understood, which is convenient for timely improvement of behavior.(3) The calculation method for dynamic satisfactions considering the correlation parameter is proposed.(4) A stable person-position matching model considering dynamic satisfactions is established.
Limitations of this paper are as follows: (1) The considered initial evaluation information is only presented in the form of hesitant fuzzy numbers, without considering other forms of fuzzy preference information.(2) Multi-attribute and multi-group factors in the process of person-position matching are not deeply studied; (3) The dynamic decision method proposed in this paper is difficult to solve the person-position matching problem involving psychological behaviors.
Future research will mainly focus on the following areas: (1) Multiple information expression tools and techniques that are more in line with the real situation need to be used study person-position matching decision, such as probabilistic hesitant fuzzy sets, hesitant language term sets and test algorithms, etc. (2) Dynamic person-position matching decision under multi-group and multi-attribute conditions can be discussed; (3) The influence of psychological behaviors of bilateral subjects on satisfaction need to be discussed, such as herd mentality, risk aversion behavior and regret behavior.(4) More complex dynamic bilateral matching problems in other fields will be explored.
and s b ij (t k ) are obtained by Eq. (1) as follows: By Eqs.(2) and (3), extended expected score matrices S

]
f ×g and matching matrix X = [x ij ] f ×g , the following multi-objective model (M-1) is established with goals of maximizing dynamic satisfactions of managers of positions and candidates and constraints of stable matching: wherein, Max D A = denotes maximizing dynamic satisfactions of candidates.

]]
f ×g of side B to side A is established by Eq. f ×g and the matching matrixX = [x ij ] f ×g , a stable person-position matching model (M−1) considering dynamic satisfactions is established.Step 10.Model (M−1) is transformed into Model (M−2) by the linear weighting method, and Model (M−2)